Problem: Find the least common multiple $(\text{LCM})$ of $5m^7+35m^6+50m^5$ and $-20m^5-80m^4+100m^3$. You can give your answer in its factored form.
The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $5m^7+35m^6+50m^5$ can be factored as ${(5)(m^3)}{(m^2)}{(m+5)}{(m+2)}$ by factoring out a $5m^5$ and using the sum-product pattern. $-20m^5-80m^4+100m^3$ can be factored as ${(2^2)}{(5)(m^3)(m+5)}{(1-m)}$ by factoring out a $20m^3$ and using the sum-product pattern. We can see that: Both polynomials share the factors ${(5)(m^3)(m+5)}$ Only the first polynomial has the factors ${(m^2)(m+2)}$ Only the second polynomial has the factors ${(2^2)(1-m)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(5)(m^3)(m+5)}{(m^2)(m+2)}{(2^2)(1-m)}\\\\ &=20(m^5)(m+2)(m+5)(1-m)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $20(m^5)(m+2)(m+5)(1-m)$.